Multi-channel frequency-domain adaptive filter method and apparatus

ABSTRACT

The invention is a method and apparatus for frequency-domain adaptive filtering that has broad applications such as to equalizers, but is particularly suitable for use in acoustic echo cancellation circuits for stereophonic and other multiple channel teleconferencing systems. The method and apparatus utilizes a frequency-domain recursive least squares criterion that minimizes the error signal in the frequency-domain. In order to reduce the complexity of the algorithm, a constraint is removed resulting in an unconstrained frequency-domain recursive least mean squares method and apparatus. A method and apparatus for selecting an optimal adaptation step for the UFLMS is disclosed. The method and apparatus is generalized to the multiple channel case and exploits the cross-power spectra among all of the channels.

FIELD OF THE INVENTION

The present invention pertains to adaptive filtering in multi-channelenvironments. More particularly, the invention has specific applicationto multi-channel acoustic echo cancellation such as in stereophonic andother multi-channel teleconferencing systems.

BACKGROUND OF THE INVENTION

The evolution of teleconferencing to a more lifelike and transparentaudio/video medium depends upon, among other things, the evolution ofteleconferencing audio capabilities. The more realistic the sound, themore lifelike a teleconference will be and the more persons andbusinesses will use it. Some present-day teleconferencing systems havealready evolved to the point of including high-fidelity audio systems(100-7000 Hz bandwidth). These systems provide a significant improvementover older telephone systems (200-3200 Hz bandwidth). However, such highfidelity systems are by no means the limits of audio evolution inteleconferencing.

Spatial realism is highly desirable for audio/video teleconferencing.This is because of the need of a listener to follow, for example, adiscussion among a panel of dynamic, multiple, and possibly simultaneoustalkers. The need for spatial realism leads to consideration ofmulti-channel audio systems in teleconferencing, which, at a minimum,involves two channels (i.e., stereophonic).

Many present-day teleconferencing systems have a single (monophonic)full-duplex audio channel for voice communication. These systems, whichrange from simple speaker-phones to modern video teleconferencingsystems, typically employ acoustic echo cancellers (AECs) to removeundesired echos that result from acoustic coupling. This couplingresults when sound emitted from the teleconference loudspeaker (inresponse to a signal from a remote location), arrives at theteleconference microphone in the same room (i.e., the echo). Themicrophone generates a signal in response to this sound that is returnedto the remote room in which it was originally generated. An AEC employsan adaptive filter to estimate the impulse response from the loudspeakerto the microphone in a room in which an echo occurs and to generate asignal to be subtracted from the receiver signal to cancel that echoelectrically. Like monophonic teleconferencing, high-qualitystereophonic teleconferencing requires AEC.

Stereophonic AEC presents a problem which does not exist in themonophonic context. In monophonic teleconferencing systems, a singleadaptive filter is used to estimate a single impulse response from theloudspeaker to the microphone in the room experiencing an echo. There isonly one impulse response to estimate because there is only oneloudspeaker and one microphone in the room. As the adaptive filterimpulse response estimate approaches the true impulse response of theroom, the difference between these responses approaches zero. Once theirdifference is very small, the effects of echo are reduced. The abilityto reduce echo is independent of the signal from the loudspeaker, sincethe real and estimated impulse responses are equal (or nearly so) andboth the room (with its real impulse response) and the adaptive filter(with its estimated impulse response) are excited by the same signal.

In multi-channel stereophonic teleconferencing systems, multiple (e.g.,two) adaptive filters are used to estimate the multiple (e.g., two)impulse responses of the room. Each adaptive filter is associated with adistinct acoustic path from a loudspeaker to a microphone in thereceiving room. Rather than being able to independently estimate theindividual impulse responses of the room, conventional stereophonic AECsystems derive impulse responses which have a combined effect ofreducing echo. This limitation on independent response derivation is dueto the fact that the AEC system can measure only a single signal permicrophone. This signal is the sum of multiple acoustic signals arrivingat a single microphone through multiple acoustic paths. Thus, the AECcannot observe the individual impulse responses of the room. The problemwith deriving impulse response estimates based on the combined effect ofreduced echo is that such combined effect does not necessarily mean thatthe actual individual impulse responses are accurately estimated. Whenindividual impulse responses are not accurately estimated, the abilityof the AEC system to be robust to changes in the acousticcharacteristics of the remote location is limited, commonly resulting inundesirable lapses in performance.

FIG. 1 presents a schematic diagram of a conventional stereophonic(two-channel) AEC system in the context of stereo teleconferencingbetween two locations. A transmission room 1 is depicted on the right ofthe figure. Transmission room 1 includes two microphones 2, 3 which areused to pick up signals from an acoustic source 4 (e.g., a speakingperson) via two acoustic paths that are characterized by the impulseresponses g1(t) and g2(t). (For clarity of presentation, all acousticpaths are assumed to include the corresponding loudspeaker and/ormicrophone responses.) Output from microphones 2, 3 are stereophonicchannel source signals x2(t) and x1(t), respectively. These stereophonicchannel source signals, x2(t) and x1(t), are then transmitted via atelecommunications network (such as a telephone or an ATM network) toloudspeakers 11, 12 in a receiving room 10 (shown on the left). Forconvenience, this direction will herein be termed the upstream directionand transmissions in the opposite direction, i.e., from room 10 to room1, will be termed the downstream direction. The terms upstream anddownstream are not intended to be limiting and have no particularconnotation other than to differentiate between two directions.Loudspeakers 11, 12 are acoustically coupled to microphone 14 inreceiving room 10 via the paths indicated with impulse responses h1(t)and h2(t). These are the paths by which acoustic echo signals arrive atmicrophone 14.

The output of the microphone 14 is signal y(t), which is a signalrepresenting acoustic signals in the receiving room impinging on themicrophone. These acoustic signals include the acoustic echo signals aswell as any signals independently generated in the room (such as by aspeaking person). Loudspeakers 11, 12 are also acoustically coupled tomicrophone 13 by other acoustic paths. For clarity of presentation,however, only the coupling to microphone 14 and AEC with respect to itsoutput will be discussed.

Further, those of ordinary skill in the art will recognize that theanalysis concerning AEC for the output of microphone 14 is applicable tothe output of microphone 13 as well. Similarly, those skilled in the artwill recognize that AEC as performed for the outputs of microphones 13and 14 in receiving room 10 also may be advantageously performed for theoutputs of microphones 2 and 3 in transmitting room 1, wherein thefunctions of receiving room 10 and transmitting room 1 are swapped.

If nothing were done to cancel the acoustic echo signals in receivingroom 10, these echoes would pass back to loudspeaker 5 in transmissionroom 1 (via microphone 14 and the telecommunications network) and wouldbe circulated repeatedly, producing undesirable multiple echoes, or evenworse, howling instability. This, of course, is the reason thatproviding AEC capability is advantageous.

Conventional AECs typically derive an estimate of the echo with use of afinite impulse response (FIR) filter with adjustable coefficients. This“adaptable” filter models the acoustic impulse response of the echo pathin the receiving room 10. FIG. 1 generally illustrates this techniquewith use of AEC 20 using two adaptive FIR filters 16, 15 having impulseresponses, ĥ1(t) and ĥ2(t), respectively, to model the two echo paths inthe receiving room 10. Filters 16, 15 may be located anywhere in thesystem (i.e., at the transmitting room 1, in the telecommunicationsnetwork, or at the receiving room 10), but are preferably located at thereceiving room 10.

Driving these filters 16, 15 with the upstream loudspeaker signals x1(t)and x2(t) produces signals ŷ1(t) and ŷ2(t), which are components of atotal echo estimate. The sum of these two echo estimate componentsignals yields the total echo estimate signal, ŷ(t), at the output ofsumming circuit 17. This echo estimate signal, ŷ(t), is subtracted fromthe downstream signal y(t) by subtraction circuit 18 to form an errorsignal e(t). Error signal e(t) is intended to be small (i.e., driventowards zero) in the absence of near-end speech (i.e., speech generatedin the receiving room).

In most conventional AEC applications, the coefficients of adaptivefilters 15, 16 are derived using well-known techniques, such as thefamiliar LMS (or stochastic gradient) algorithm. The coefficients areupdated in an effort to reduce the error signal to zero. As such, thecoefficients ĥ1(t) and ĥ2(t) are a function of the stereophonic signalsx2(t) and x1(t) and the error signal, e(t).

As mentioned above, unlike monophonic AECs, conventional stereophonicAECs do not independently estimate the individual impulse responses of aroom. Rather, conventional stereophonic AEC systems derive impulseresponses which have a combined effect of reducing echo. Unlessindividual impulse responses are accurately estimated, the ability ofthe AEC system to be robust to changes in the acoustic characteristicsof the remote location is limited and undesirable lapses in performancemay occur.

This problem is discussed fully in U.S. patent application Ser. No.09/395,834, entitled A Frequency Domain Stereophonic Acoustic EchoCanceller Utilizing Non-Linear Transformation, which is incorporatedherein by reference.

Not only must the adaptation algorithm of filters 15, 16 trackvariations in the receiving room, it must also track variations in thetransmission room. The latter variations are particularly difficult totrack. For instance, if one talker stops talking and another startstalking at a different location in the room, the impulse responses, g1and g2, change abruptly and by very large amounts.

J. Benesty, A. Gilloire, Y. Grenier, A frequency domain stereophonicacoustic echo canceller exploiting the coherence between the channels,Acoustic Research Letters Online, 21 Jul. 1999, discloses a frequencydomain algorithm for use in a stereophonic echo canceller that exploitsthe coherence between the channels.

As can be seen from the above discussion, therefore, the challenge is todevise an approach which (as in the case of a single-channel echocanceller) converges independently of variations in the transmissionroom. Thus, it is desirable to de-correlate x1 and x2.

U.S. Pat. No. 6,694,020 discloses a teleconferencing system thatde-correlates the channel signals in a multi-channel teleconferencingsystem. FIG. 2 is a schematic diagram of a stereophonic teleconferencingsystem that includes circuitry for de-correlating x₁ and x₂ inaccordance with the teachings of U.S. Pat. No. 5,828,756.

The system of FIG. 2 is identical to that of FIG. 1 except for thepresence of non-linear signal transformation modules 25, 30 (NL), whichhave been inserted in the paths between microphones 3, 2 of transmissionroom 1 and loudspeakers 11, 12 of receiving room 10. By operation ofnon-linear transformation modules 25, 30, stereophonic source signalsx₁(t) and x₂(t) are transformed to signals x₁′(t) and x₂′(t),respectively, where “′” indicates a transformed signal which (in thiscase) advantageously has a reduced correlation with the othertransformed signal of the stereophonic system.

As with the system presented in FIG. 1, the filters of AEC 20 may belocated anywhere within the system, but are preferably located atreceiving room 10. Non-linear transformation modules 25, 30 also may belocated anywhere (so long as receiving room 10 and AEC 20 both receivethe transformed signals as shown), but are preferably located attransmitting room 1.

Specifically, in accordance with one embodiment of the device disclosedin U.S. Pat. No. 5,828,756, the signals x₁(t) and x₂(t) areadvantageously partially de-correlated by adding to each a smallnon-linear function of the corresponding signal itself. It is well-knownthat the coherence magnitude between two processes is equal to one (1)if and only if they are linearly dependent. Therefore, by adding a“noise” component to each signal, the coherence is reduced. However, bycombining the signal with an additive component which is similar to theoriginal signal, the audible degradation may be advantageouslyminimized, as compared to the effect of adding, for example, a randomnoise component. This is particularly true for signals such as speech,where the harmonic structure of the signal tends to mask the distortion.

Thus, a linear relation between x₁′(t) and x₂′(t) is avoided, therebyensuring that the coherence magnitude will be smaller than one. Such atransformation reduces the coherence and hence the condition number ofthe covariance matrix, thereby improving the misalignment. Of course,the use of this transformation is particularly advantageous when itsinfluence is inaudible and does not have any deleterious effect onstereo perception. For this reason, it is preferable that themultiplier, α, be relatively small.

In one illustrative embodiment of U.S. Pat. No. 5,828,756, thenon-linear functions f₁ and f₂ as applied by non-linear function module32 are each half-wave rectifier functions.

The above-described solution proposed in U.S. Pat. No. 5,828,756 is asimple and efficient solution that overcomes the above-discussedproblems by adding a small non-linearity into each channel. Thedistortion due to the non-linearity is hardly perceptible and does notaffect the stereo effect, yet reduces inter-channel coherence, therebyallowing reduction of misalignment to a low level. However, because theintroduced distortion is so small (so as not to significantly affectsound quality), the echo cancellation algorithm must be very powerful inorder to converge to a solution within a reasonably small period of timewhen conditions in the room change. A least mean squares (LMS) solutiondoes not converge fast enough. A much more powerful algorithm isnecessary in order to make the system illustrated in FIG. 2 work.

The aforementioned U.S. Pat. No. 6,694,020 discloses an acoustic echocanceller that exploits the coherence between multiple channels in thesystem illustrated in FIG. 2. Particularly, it discloses one efficientfrequency-domain adaptive algorithm used in the echo canceller circuits.

SUMMARY OF THE INVENTION

The invention is a multiple channel adaptive filtering method andapparatus that is particularly suitable for use for acoustic echocancellation in a multi-channel teleconferencing system, but has muchbroader application to any type of adaptive filtering environment. Themethod and apparatus exploits a new frequency-domain adaptive algorithmby using a frequency-domain recursive least squares criterion thatminimizes the error signal in the frequency-domain. In accordance withthe method and apparatus, an exact adaptive algorithm based on thenormal equation is provided. Further, in order to reduce the complexityof the algorithm, a constraint is removed resulting in an unconstrainedfrequency-domain least mean squares (UFLMS) algorithm.

The invention further includes a method and apparatus for selecting anoptimal step size for the UFLMS. Most importantly, the algorithm isgeneralized to the multiple channel case thereby exploiting thecross-power spectra among all the channels which allows for a fastconvergence rate in the multiple channel acoustic echo cancellationenvironment where the input signals are highly correlated.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is a schematic diagram of a conventional stereophonicteleconferencing system.

FIG. 2 is a schematic diagram of a prior art stereophonicteleconferencing system in which small non-linearities have been addedinto the channel paths in order to selectively reduce the correlationbetween the individual channel signals.

FIG. 3 is a schematic diagram in accordance with an illustrativeembodiment of the present invention.

FIG. 4A is a graphical diagram illustrating convergence of the meansquare error (MSE) of a stereophonic teleconferencing system inaccordance with a normalized least mean-squares (NLMS) echo cancellationalgorithm of the prior art.

FIG. 4B is a graphical diagram illustrating misalignment in astereophonic teleconferencing system employing echo cancellation inaccordance with an NLMS echo cancellation algorithm of the prior art.

FIG. 5A is a graphical diagram illustrating convergence of the meansquare error (MSE) of a stereophonic teleconferencing system inaccordance with an FLRS echo cancellation algorithm of the prior art.

FIG. 5B is a graphical diagram illustrating misalignment in astereophonic teleconferencing system employing echo cancellation inaccordance with an FRLS echo cancellation algorithm of the prior art.

FIG. 6A is a graphical diagram illustrating convergence of the meansquare error (MSE) of a stereophonic teleconferencing system inaccordance with a first embodiment of the present invention.

FIG. 6B is a graphical diagram illustrating misalignment in astereophonic teleconferencing system employing echo cancellation inaccordance with the first embodiment of the present invention.

FIG. 7A is a graphical diagram illustrating convergence of the meansquare error (MSE) in a stereophonic teleconferencing system inaccordance with a second embodiment of the present invention.

FIG. 7B is a graphical diagram illustrating misalignment in astereophonic teleconferencing system in accordance with the secondembodiment of the present invention.

DETAILED DESCRIPTION OF THE INVENTION

FIG. 3 is a schematic diagram of a stereophonic teleconferencing systemin accordance with the present invention. It is essentially identical toFIG. 2 except for the fact that the adaptive filters 15 and 16 have beenreplaced with adaptive filters 302 and 304. As will be explained indetail below, adaptive filters 302 and 304 implement a frequency domainadaptive algorithm that tracks variations in the receiving room impulseresponses h1, h2, respectively.

1. INTRODUCTION

Since its first introduction by Dentino et al. [1], adaptive filteringin the frequency-domain has progressed very fast, and differentsophisticated algorithms have since been proposed. Ferrara [2] was thefirst to elaborate an efficient frequency-domain least mean squareadaptive filter algorithm (FLMS) that converges to the optimal (Wiener)solution. Mansour and Gray [3] derived an even more efficient algorithm,the unconstrained FLMS (UFLMS), using only three FFT operations perblock instead of five for the FLMS with comparable performances [4].However, a major handicap of these solutions is delay. Indeed, thisdelay is equal to the length of the adaptive filter L, which isconsiderable for some applications like acoustic echo cancellation (AEC)where the number of taps can easily exceed one thousand.

A new structure, using the classical overlap save (OLS) method, wasproposed in [5] and [6] and generalized in [7] where the blockprocessing N was made independent of the filter length L and N can bechosen as small as desired, with a delay equal to N. Although from thestandpoint of complexity, the optimal choice is N=L, using smaller blocksizes (N<L) in order to reduce the delay is still more efficient thanthe time-domain algorithms. A more general scheme based on weightedoverlap and add (WOLA) methods, the generalized filter (GMDF), wasproposed in [8], [9], where is the overlap factor. The settings α>1appear to be very useful in the context of adaptive filtering, since thefilter coefficients can be adapted more frequently (every N/α samplesinstead of every N samples in the standard OLS scheme). So thisstructure introduces one more degree of freedom, but the complexity isincreased by a factor of roughly α. Using a block size as large as thedelay permits will increase the convergence rate of the algorithm, whiletaking an overlap factor greater than 1 will increase the trackingabilities of the algorithm.

A new frequency-domain adaptive algorithm is derived using afrequency-domain recursive least squares criterion. A similar criterionwas proposed in [3] using mathematical expectations instead. Here,however, an exact adaptive algorithm is derived from the normalequation. The obtained algorithm is complex to implement. To reduce thecomplexity, a constraint is removed that will render exactly the UFLMS[3]. A scheme for selecting the optimal adaptation step for UFLMS isdisclosed. Most importantly, we generalize all this to the multi-channelcase. The obtained algorithm exploits the cross-power spectra among allthe channels, which is very important (for a fast convergence rate) inmulti-channel AEC where the input signals are highly correlated [10]. Tosimplify the presentation, we will derive all of the algorithms only forL=N, α=1, and with the OLS method. Generalization to any other case isstraightforward.

In the equations below the following conventions should be noted: (1)lower case bold face variables denote vectors, (2) upper case bold facevariables denote matrices, (3) underlined variables denotefrequency-domain values, and (4) variables with hats denote estimates.

2. MONO-CHANNEL FREQUENCY-DOMAIN ADAPTIVE FILTERING RE-VISITED

In the time-domain, the general procedure to derive an adaptivealgorithm is to first define an error signal, then to build a costfunction based on the error signal, and finally to minimize the costfunction with respect to the adaptive filter coefficients [11]. In thecontext of system identification, the error signal at time n between thesystem and model filter outputs is given bye(n)=y(n)−(n)  Eq. (1)whereŷ(n)=ĥ ^(T)×(n)  Eq. (2)is an estimate of the output signal y(n),ĥ=[ĥo ĥ₁ . . . ĥ_(L-1)]^(T)is the model filter, and

${x(n)} = \underset{15}{\lbrack {{x(n)}\mspace{11mu}{x( {n - 1} )}\mspace{14mu}\ldots\mspace{20mu}{x( {n - L + 1} )}} \rbrack^{T}}$is a vector containing the last L samples of the input signal x.Superscript ^(T) denotes the transpose of a vector or matrix. Therecursive least squares (RLS) adaptive algorithm is obtained exactlyfrom the normal equation which is derived by minimizing the followingtime-domain criterion [11]:

$\begin{matrix}{{{J_{t}(n)} = {( {1 - \lambda_{t}} ){\sum\limits_{p = 0}^{n}{\lambda_{t}^{n - p}{{\mathbb{e}}^{2}(p)}}}}},} & {{Eq}.\mspace{11mu}(3)}\end{matrix}$where λ_(t) (0<λt<1) is an exponential forgetting factor. In the rest ofthis section, we will follow the same approach.

We now define the block error signal (of length N=L) as:e(m)=y(m)−ŷ(m),  Eq. (4)

where m is the block time index, and

e(m) = [e(m L)  …  e(m L + L − 1)]^(T), y(m) = [y(m L)  …  y(m L + L − 1)]^(T), ŷ(m) = [x(m L)  …  x(m L + L − 1]^(T)ĥ = X^(T)(m)ĥ.It can easily be checked that X is a Toeplitz matrix of size (L×L).

It is well known that a Toeplitz matrix X can be transformed, bydoubling its size, to a circulant matrix

$c = \begin{bmatrix}x^{\prime} & x \\x & x^{\prime}\end{bmatrix}$where X′ also is a Toeplitz matrix. (The matrix X′ can be expressed interms of the elements of X, except for an arbitrary diagonal.) Usingcirculant matrices, the block error signal can be re-writtenequivalently:

$\begin{matrix}{{\begin{bmatrix}0_{{Lx}\; 1} \\{e(m)}\end{bmatrix} = {\begin{bmatrix}0_{{Lx}\; 1} \\{y(m)}\end{bmatrix} - {W{{\hat{y}}^{\prime}(m)}}}},} & {{Eq}.\mspace{14mu}(5)} \\{where} & \; \\{W = \begin{bmatrix}0_{LxL} & 0_{LXL} \\0_{LxL} & I_{LxL}\end{bmatrix}} & \; \\{and} & \; \\{{{{\hat{y}}^{\prime}(m)} = {{c(m)}\begin{bmatrix}\hat{h} \\o_{{Lx}\; 1}\end{bmatrix}}},} & {{Eq}.\mspace{14mu}(6)}\end{matrix}$

It is also well known that a circulant matrix is easily decomposed asfollows:C=F− ¹ DF,where F is the Fourier matrix of size (2 L×2 L) and D is a diagonalmatrix whose elements are the discrete Fourier transform of the firstcolumn of C. If we multiply Eq. (5) by F, we get the error signal in thefrequency domain:

$\begin{matrix}\begin{matrix}{{\underset{\_}{e}(m)} = {{\underset{\_}{y}(m)} - {G{\underset{\_}{{\hat{y}}^{\prime}}(m)}}}} \\{{= {{\underset{\_}{y}(m)} - {{{GD}(m)}\hat{\underset{\_}{h}}}}},}\end{matrix} & {{Eq}.\mspace{14mu}(7)} \\{where} & \; \\{{{{\underset{\_}{e}(m)} = {F\begin{bmatrix}0_{{Lx}\; 1} \\{e(m)}\end{bmatrix}}},{{\underset{\_}{y}(m)} = {F\begin{bmatrix}0_{{Lx}\; 1} \\{y(m)}\end{bmatrix}}},{G = {FWF}^{- 1}},\lbrack {{{{\hat{y}}^{\prime}(m)} = {F{{\hat{y}}^{\prime}(m)}}},} \rbrack}{{{\underset{\_}{{\hat{y}}^{\prime}}(m)} = {F{{\hat{y}}^{\prime}(m)}}},{\hat{\underset{\_}{h}} = {F\begin{bmatrix}\hat{h} \\0_{{Lx}\; 1}\end{bmatrix}}},}} & \;\end{matrix}$

Having derived a frequency-domain error signal, we now define afrequency-domain criterion which is similar to Eq. (3):

$\begin{matrix}{{{J_{f}(m)} = {( {1 - \lambda_{f}} ){\sum\limits_{p = 0}^{m}\;{\lambda_{f}^{m - p}{{\underset{\_}{e}}^{H}(p)}{\underset{\_}{e}(p)}}}}},} & {{Eq}.\mspace{14mu}(8)}\end{matrix}$

where ^(H) denotes conjugate transpose. Let ∇ be the gradient operator(with respect to ĥ). Applying the operator ∇ to the cost function J_(f),we obtain (noting that G^(H)G=G²=G) the complex gradient vector:

$\begin{matrix}\begin{matrix}{{\nabla\;{J_{f}(m)}} = \frac{\partial{J_{f}(m)}}{\partial{\hat{\underset{\_}{h}}(m)}}} \\{= {{{- ( {1 - \lambda_{f}} )}{\sum\limits_{p = 0}^{m}\;{\lambda_{f}^{m - p}{D(p)}G^{*}{{\underset{\_}{y}}^{*}(p)}}}} +}} \\{{{( {1 - \lambda_{f}} )\lbrack {\sum\limits_{p = 0}^{m}\;{\lambda_{f}^{m - p}{D(p)}G^{*}{D^{*}(p)}}} \rbrack}{{\hat{\underset{\_}{h}}}^{*}(m)}},}\end{matrix} & {{Eq}.\mspace{14mu}(9)}\end{matrix}$where * denotes complex conjugate. By setting the gradient of the costfunction equal to zero, conjugating, and noting that Gy(p)=y(p), weobtain the so-called normal equation:

$\begin{matrix}{{{{S(m)}{\hat{\underset{\_}{h}}(m)}} = {s(m)}},} & {{Eq}.\mspace{14mu}(10)} \\{where} & \; \\\begin{matrix}{{S(m)} = {( {1 - \lambda_{f}} ){\sum\limits_{p = 0}^{m}\;{\lambda_{f}^{m - p}{D^{*}(p)}{{GD}(p)}}}}} \\{= {{\lambda_{f}{S( {m - 1} )}} + {( {1 - \lambda_{f}} ){D^{*}(m)}{{GD}(m)}}}}\end{matrix} & {{Eq}.\mspace{14mu}(11)} \\{and} & \; \\\begin{matrix}{{s(m)} = {( {1 - \lambda_{f}} ){\sum\limits_{p = 0}^{m}\;{\lambda_{f}^{m - p}{D^{*}(p)}{\underset{\_}{y}(p)}}}}} \\{= {{\lambda_{f}{s( {m - 1} )}} + {( {1 - \lambda_{f}} ){D^{*}(m)}{{\underset{\_}{y}(m)}.}}}}\end{matrix} & {{Eq}.\mspace{14mu}(12)}\end{matrix}$

It can be shown that, if the covariance matrix of the input signal is ofrank L, then the matrix S(m) is nonsingular [3], [4]. In this case, thenormal equation has a unique solution which is the optimal Wienersolution.

Enforcing the normal equation at block time indices m and m−1, and usingEq. (11) and Eq. (12), we easily derive an exact adaptive algorithm:e (m)= y (m)−GD(m) ĥ (m−1)  Eq. (13)ĥ (m)= ĥ (m−1)+(1−λ_(f))S ⁻¹(m)D*(m) e (m).  Eq. (14)This will be called the constrained algorithm. Note that this definitionis different from the original frequency-domain adaptive algorithmproposed by Ferrara [2]. (The constraint here is on the update of thematrix S while, in Ferrara's algorithm, the constraint is on the updateof the coefficients of the filter). It can be shown that the convergenceof the proposed algorithm for stationary signals does not depend on thestatistics of the input signal which is, of course, a very desirablefeature.

Frequency-domain adaptive algorithms were first introduced to reduce thearithmetic complexity of the LMS algorithm [2]. Unfortunately, thematrix S is not diagonal, so the proposed algorithm has a highcomplexity and may not be practical to implement. If, however, 2 G canbe well approximated by the identity matrix, we then obtain thefollowing unconstrained algorithm.S _(u)(m)=λ_(f) S _(u)(m−1)+(1−λ_(f))D*(m)D*(m)  Eq. (15)ĥ (m)= ĥ (m−1)+μ_(u) S _(u) ⁻¹(m)D*(m) e (m)  Eq. (16)where S_(u) is now a diagonal matrix and μ_(u)=2(1−λ_(f)) is a positivenumber. This algorithm is exactly the unconstrained frequency-domainadaptive filter proposed by Mansour and Gray [3] and since S_(u) isdiagonal, this algorithm is very attractive from a complexity point ofview. Below, it is shown that this approximation is justified. Also,below we disclose the optimum value for μ_(u).

Let us examine the structure of the matrix G. We have: G*=F−¹WF; since Wis a diagonal matrix, and G is a circulant matrix. Therefore, inversetransforming the diagonal of W gives the first column of G*,

$\begin{matrix}{g^{*} = \lbrack {g_{0}^{*}g_{1}^{*}\mspace{14mu}\ldots\mspace{14mu} g_{{2\; L} - 1}^{*}} \rbrack^{T}} \\{= {F^{- 1}{\lceil {0\mspace{14mu}\ldots\mspace{14mu} 0\mspace{14mu} 1\mspace{14mu}\ldots\mspace{14mu} 1} \rceil^{T}.}}}\end{matrix}$The elements of vector g can be written explicitly as:

$\begin{matrix}\begin{matrix}{g_{k} = {\frac{1}{2\; L}{\sum\limits_{l = L}^{{2\; L} - 1}\;{\exp( {{- {\mathbb{i}2}}\;\pi\;{{kl}/2}\; L} )}}}} \\{{= {\frac{( {- 1} )^{k}}{2\; L}{\sum\limits_{l = 0}^{L - 1}\;{\exp\lbrack {{- {\mathbb{i}}}\;\pi\;{{kl}/L}} \rbrack}}}},}\end{matrix} & {{Eq}.\mspace{14mu}(17)}\end{matrix}$where i²=−1. Since g_(k) is the sum of a geometric progression, we have:

$\begin{matrix}\begin{matrix}{{gk} = \{ \begin{matrix}0.5 & {k = 0} \\{\frac{( {- 1} )^{k}}{2\; L}\frac{1 - {\exp( {{- {\mathbb{i}}}\;\pi\; k} )}}{1 - {\exp( {{- {\mathbb{i}}}\;\pi\;{k/L}} )}}} & {k \neq 0}\end{matrix} } \\{= \{ \begin{matrix}0.5 & {k = 0} \\0 & {k\mspace{14mu}{even}} \\{\frac{- 1}{2\; L}\lbrack {1 - {i\;{\cot( \frac{\pi\; k}{2\; L} )}}} \rbrack} & {{k\mspace{14mu}{odd}},}\end{matrix} }\end{matrix} & {{Eq}.\mspace{14mu}(18)}\end{matrix}$where L−1 elements of vector g are equal to zero. Moreover, sinceG^(H)G=G, then g^(H) g=g₀=0.5 and we have

$\begin{matrix}{{{g^{H}g} - g_{0}^{2}} = {{\sum\limits_{l = 1}^{{2\; L} - 1}\;{{gl}}^{2}} = {{2{\sum\limits_{l = 1}^{L - 1}\;{{gl}}^{2}}} = \frac{1}{4}}}} & {{Eq}.\mspace{14mu}(19)}\end{matrix}$

We can see from Eq. (19) that the first element of vector g, i.e., g₀,is dominant in a mean-square sense, and from Eq. (18) that the first Lelements of g decrease rapidly to zero as k increases. Because of theconjugate symmetry, some of the last elements of g are non-negligible.However, this is of little concern since G is circulant with g as itsfirst column and its other columns have those non-negligible elementsshifted in such a way that they are concentrated around the maindiagonal.

To summarize, only the first few off-diagonals of G will benon-negligible, while the others can be completely neglected. Thus,approximating G by a diagonal matrix, i.e., G^({tilde over ( )})g0I=I/2, is reasonable, and in this case we will haveλu ^({tilde over ( )})(1−λ_(f))/g ₀=2(1−λ_(f))for an optimal convergence rate. Note that this is in agreement withprevious derivations [2] that give an optimal step size of 1−λ_(f)divided by a power normalizing factor, which for our assumed unit-powersignal with L-sample zero padding is equal to ½.

3. GENERALIZATION TO THE MULTI-CHANNEL CASE

The generalization to the multi-channel case is rather straightforward.Therefore, this section only highlights some important steps and statesthe algorithms. For convenience, we will use the same notation aspreviously employed. Let J be the number of channels. Our definition ofmulti-channel is that we have a system with J input signals x_(j), j=1,2 . . . , J and one output signal y. Now the block error signal isdefined as:

$\begin{matrix}{{{e(m)} = {{y(m)} - {\sum\limits_{j = 1}^{J}\;{{X_{j}^{T}(m)}{\hat{h}}_{j}}}}},} & {{Eq}.\mspace{14mu}(20)}\end{matrix}$

where e and y are vectors of e_(j) and y_(j) respectively, all matricesX_(j) are Toeplitz of size (L×L), and ĥ_(j) is the estimated impulseresponse of the jth channel. In the frequency-domain, we have [c.f. Eq.(7)]:

$\begin{matrix}\begin{matrix}{{\underset{\_}{e}(m)} = {{\underset{\_}{y}(m)} - {G{\sum\limits_{j = 1}^{J}\;{{D_{j}(m)}{\hat{\underset{\_}{h}}}_{j}}}}}} \\{{= {{\underset{\_}{y}(m)} - {{{GD}(m)}\hat{\underset{\_}{h}}}}},}\end{matrix} & {{Eq}.\mspace{14mu}(21)}\end{matrix}$where D=[D₁D₂ . . . D_(J)] is a (2 L×2 LJ) matrix containing all the Jdiagonal matrices Dj and ĥ=[ĥ ₁ ^(T) ĥ ₂ ^(T) . . . ĥ _(J) ^(T)]^(T) isthe (2 LJ×1) vector of concatenated, transformed, zero-padded estimatedimpulse responses. Minimizing the criterion defined in Eq. (8), weobtain the normal equation for the multi-channel case:

$\begin{matrix}{{{S(m)}{\hat{\underset{\_}{h}}(m)}} = {s(m)}} & {{Eq}.\mspace{14mu}(22)} \\\begin{matrix}{{S(m)} = {( {1 - \lambda_{f}} ){\sum\limits_{p = 0}^{m}\;{\lambda_{f}^{m - p}{D^{H}(p)}{\underset{\_}{y}(p)}}}}} \\{= {{\lambda_{f}{s( {m - 1} )}} + {( {1 - \lambda_{f}} ){D^{H}(m)}{\underset{\_}{y}(m)}}}}\end{matrix} & {{Eq}.\mspace{14mu}(23)}\end{matrix}$is a (2 LJ×2 LJ) matrix and

$\begin{matrix}\begin{matrix}{{S(m)} = {( {1 - \lambda_{f}} ){\sum\limits_{p = 0}^{m}\;{\lambda_{f}^{m - p}{D^{H}(p)}{{GD}(p)}}}}} \\{= {{\lambda_{f}{S( {m - 1} )}} + {( {1 - \lambda_{f}} ){D^{H}(m)}{{GD}(m)}}}}\end{matrix} & {{Eq}.\mspace{14mu}(24)}\end{matrix}$is a (2 LJ×1) vector. Using the same approach and definitions as inSection 2, we obtain the multi-channel, constrained, frequency-domain,adaptive algorithm:e (m)= y (m)−GD(m) ĥ (m−1)  Eq. (25)ĥ (m)= ĥ (m−1)+(1−λ_(f))S ⁻¹(m)D ^(H)(m) e (m)  Eq. (26)and the multi-channel unconstrained frequency-domain adaptive algorithm:S _(u)(m)=λ_(f) S _(u)(m−1)+(1−λ_(f))D ^(H)(m)D(m)  Eq. (27)ĥ (m)= ĥ (m−1)+μ_(u) S _(u) ⁻¹(m)D ^(H)(m) e (m)  Eq. (28)Now, S_(u) is not a diagonal matrix, but a block matrix containing J²diagonal matrices that are estimates of the power spectra andcross-power spectra of all the input signals.Particular case: The two-channel unconstrained frequency-domain adaptivealgorithm.

We easily deduce the algorithm from Eq. (27) and Eq. (28):e (m)= y (m)−G[D ₁(m) ĥ ₁(m−1)+D ₂(m) ĥ ₂(m−1)]  Eq. (29)ĥ ₁(m)= ĥ ₁(m−1)+μ_(u) S ₁ ⁻¹(m)[D ₁*(m)−S _(1,2)(m)S _(2,2) ⁻¹(m)D₂*(m)] e (m)  Eq. (30)ĥ ₂(m)= ĥ ₂(m−1)+μ_(u) S ₂ ⁻¹(m)[D ₂*(m)−S _(2,1)(m)S _(1,1) ⁻¹(m)D₁*(m)] e (m)  Eq (31)where S_(j,l) are the (diagonal) sub-matrices of S_(u),S _(j)(m)=S _(j,j)(m)[I _(2L×2L)−Γ^(H)(m)Γ(m)],j=1,2  Eq. (32)andΓ(m)=[S _(1,1)(m)S _(2,2)(m)]^(−1/2) S _(1,2)(m)  Eq. (33)is the coherence matrix. This algorithm is exactly the same as in [13],exploiting the coherence between the two channels in order to improvethe convergence rate of the adaptive filter.

For simplicity, we have derived the multi-channel case assuming N=L andno overlap (α=1). It is easy to generalize this for α>1 by simplycomputing the FFTs using overlapped data, which is exactly what is donein the next section for the application example (α=4). Furthermore, itis straightforward, although tedious, to generalize to the case of N<L[8].

4. APPLICATION TO ACOUSTIC ECHO CANCELLATION AND SIMULATIONS

Multi-channel acoustic echo cancellation (AEC) is typically intended foruse in high quality teleconferencing systems and in multi participantdesktop conferencing, implementing sound transmission through at leasttwo channels. Multi-channel AEC can be viewed as a simple generalizationof the single-channel acoustic echo cancellation principle. FIGS. 1, 2and 3 show this technique, in the two-channel (stereo) case, for onemicrophone in the receiving room (which is represented by the two echopaths h₁ and h₂ between the two loudspeakers and the microphone). Thetwo reference signals x₁ and x₂ from the transmission room are obtainedfrom two microphones in the case of teleconferencing. These signals arederived by filtering from a common source and this gives rise to thenon-uniqueness problem discussed in the Background section that does notarise for the single-channel AEC. Also, as previously noted, the usualadaptive algorithms, therefore, converge to solutions that depend on theimpulse responses in the transmission room. This means that for goodecho cancellation one must track not only the changes in the receivingroom, but also the changes in the transmission room (for example, whenone person stops talking and another person starts). The same problemoccurs for multi-channel desktop conferencing, where multi-channel soundis synthesized from the single-microphone signals of all theparticipants [12].

Also as discussed in the Background section, U.S. Pat. No. 5,828,756proposes a simple but efficient solution that overcomes the above-notedproblem by adding a small non-linearity into each channel (FIGS. 2 and3). The distortion due to the non-linearity is hardly perceptible forspeech yet it reduces inter-channel coherence thereby allowing reductionof misalignment to a low level. However, this solution is fruitful onlywhen combined with the multi-channel FRLS algorithm which implies a highlevel of computational complexity such that a real time implementationis difficult. Aforementioned U.S. Pat. No. 6,696,020 discloses analternate, frequency-domain based, algorithm for generating the acousticecho cancellation signal based on an extended least mean squares (ELMS)scheme. The present invention is an alternative efficient multi-channelfrequency-domain adaptive method and apparatus that is even lesscomputationally burdensome.

We now show, by way of simulation using two channels, that the proposedunconstrained frequency-domain adaptive filter is a good alternative tosome classical algorithms, namely, the two-channel Normalized Least MeanSquares (NLMS) and the two-channel Fast Recursive Least Squares (FRLS).The signal source s in the transmission room is a 10 s speech signal.The two microphone signals x₁ and x₂ were obtained by convolving s withtwo impulse responses g₁, g₂ of length 4096, as measured in an actualroom. The microphone output signal y in the receiving room is obtainedby summing the two convolutions (h₁*x₁) and h₂*x₂), where h₁ and h₂ alsowere measured in an actual room as 4096-point responses. A white noisesignal with 45 dB SNR is added to y. The sampling rate is 16 kHz. Thelength of the two adaptive filters is taken as L=1024. In all of thesimulations, we added a half-wave rectifier non linearity (with gain of0.5) to the signals x₁ and x₂. For the proposed algorithm, we used thefollowing parameters: N=1024, and α=4 (which implies an overall delayequal to 1024 samples, i.e., 64 ms). With these values of N and, theproposed algorithm is 7 times less complex than the two-channel NLMS and40 times less complex than the two-channel FRLS.

FIGS. 4A, 5A and 6A show the convergence of the Mean Square Error (MSE)for the prior art two channel Normalized Least Mean Squares (NLMS)solution, the prior art two-channel Fast Recursive Least Squares (FRLS)solution and the solution of the present invention, respectively.

FIGS. 4B, 5B and 6B show the misalignment for the same three cases,respectively.

For the purpose of smoothing the curves, error and misalignment sampleswere averaged over 128 points. For the FRLS algorithm, we chose a valueλRLS=1−1/(20 L) where L=1024 in this case. Accordingly, for the blockfrequency-domain algorithm, we chose λf=λLRLS=0.95 in order to have thesame effective window length.

It can be seen that the proposed algorithm outperforms the other twowith respect to misalignment while also being much less complex toimplement. Also, the steady-state attenuation of the MSE for thefrequency-domain algorithm appears to be as good as that for the FRLS(and somewhat better than NLMS) as confirmed by informal listeningtests. However, the initial convergence rate of the proposed algorithmis somewhat slower than that of the FRLS. This can be improved by usinga small exponential forgetting factor as shown in FIGS. 7A and 7B whichshow the convergence of the MSE and the misalignment, respectively, forthe present invention with λ_(f)=0.9, but the misalignment is thenincreased somewhat. The optimal tradeoff between convergence rate andmisalignment is very subjective and application dependent.

For clarity of explanation, the illustrative embodiments of the presentinvention described herein were presented as comprising individualfunctional blocks. The functions that these blocks represent may beprovided through the use of either shared or dedicated hardware,including, but not limited to, hardware capable of executing software.For example, the functions of the blocks presented in the variousillustrative figures may be provided by a single shared processor. (Useof the term “processor” should not be construed to refer exclusively tohardware capable of executing software.) Potential embodiments maycomprise digital signal processor (DSP) hardware, read-only memory (ROM)for storing software performing the operations discussed above, andrandom access memory (RAM) for storing DSP results. Very large scaleintegration (VLSI) hardware embodiments, as well as custom VLSIcircuitry in combination with a general purpose DSP circuit, may also beprovided.

5. CONCLUSIONS

We have derived a class of multi-channel frequency-domain adaptivealgorithms from a frequency-domain recursive least squares criterion.The constrained algorithm was deduced directly and exactly from thenormal equation and, in this sense, is optimal, while the unconstrainedversion is a good approximation. Most importantly, both algorithmsexploit the cross-power spectra (or equivalently the cross-correlationsin the time-domain) among all the channels and this feature isfundamental for the algorithms to converge rapidly to the Wienersolution, especially for applications like multi-channel AEC where thechannels are highly correlated.

While specific embodiments of the invention have been described inconnection with acoustic echo cancellation in a multiple channelteleconferencing system, it should be understood that this is merely anexemplary application and that the invention has much broaderapplicability. The invention, for instance, can be used in connectionwith virtually any multiple channel adaptive filtering application, suchas multi-channel equalizers.

Having thus described a few particular embodiments of the invention,various alterations, modifications, and improvements will readily occurto those skilled in the art. Such alterations, modifications andimprovements as are made obvious by this disclosure are intended to bepart of this description though not expressly stated herein, and areintended to be within the spirit and scope of the invention.Accordingly, the foregoing description is by way of example only, andnot limiting. The invention is limited only as defined in the followingclaims and equivalents thereto.

REFERENCES

-   [1] M. Dentino, J. McCool, and B. Widrow, “Adaptive filtering in the    frequency domain,” Proc. IEEE, vol. 66, pp. 1685-1659, December    1978.-   [2] E. R. Ferrara, Jr., “Fast implementation of LMS adaptive    filter,” IEEE Trans. Acoust., Speech, Signal Processing, vol.    ASSP-28, pp. 474-475, August 1980.-   [3] D. Mansour and A. H. Gray, J R., “Unconstrained frequency-domain    adaptive filter,” IEEE Trans. Acoust, Speech Signal Processing, vol.    ASSP-30, pp. 726-734, October 1982.-   [4] J. C. Lee and C. K. Un, “Performance analysis of    frequency-domain block LMS adaptive digital filters,” IEEE Trans.    Circuits Syst., vol. CAS-36, pp. 173-189, February 1989.-   [5] J. S. Soo and K. K. Pang, “Multidelay block frequency domain    adaptive filter,” IEEE Trans. Acoust., Speech, Signal Processing,    vol. ASSP-38, pp. 373-376, February 1990.-   [6] J. Benesty and P. Duhamel, “Fast constant modulus adaptive    algorithm,” IEE Proc.-F, vol. 138, pp. 379-387, August 1991.-   [7] J. Benesty and P. Duhamel, “A fast exact least mean square    adaptive algorithm,” IEEE Trans. Signal Processing, vol. 40, no. 12,    pp. 2904-2920, December 1992.-   [8] E. Moulines, O. Ait Amrane, and Y. Grenier, “The generalized    multidelay adaptive filter: structure and convergence analysis,”    IEEE Trans. Signal Processing, vol. 43, pp. 14-28, January 1995.-   [9] J. Prado and E. Moulines, “Frequency-domain adaptive filtering    with applications to acoustic echo cancellation,” Ann. Télécommun.,    vol. 49, pp. 414-428, 1994.-   [10] J. Benesty, F. Amand, A. Gilloire, and Y. Grenier, “Adaptive    filtering algorithms for stereophonic acoustic echo cancellation,”    in Proc. IEEE ICASSP, 1995, pp. 3099-3102.-   [11] M. G. Bellanger, Adaptive Digital Filters and Signal Analysis.    Marcel Dekker, 1987.-   [12] J. Benesty, D. R. Morgan, J. L. Hall, and M. M. Sondhi,    “Synthesized stereo combined with acoustic echo cancellation for    desktop conferencing,” Bell Labs Tech. J., vol. 3, pp. 148-158,    July-September 1998.-   [13] J. Benesty, A. Gilloire, and Y. Grenier, “A frequency-domain    stereophonic acoustic echo canceler exploiting the coherence between    the channels and using nonlinear transformations” in Proc. IWAENC,    1999.

1. A adaptive-filter-implemented method for adaptively filtering asignal comprising multiple echo responses, wherein each of said multipleecho responses corresponds to a different reception channel, said methodcomprising the steps of: (a) generating, by the adaptive filter, anestimate of an echo response corresponding to each of said multiple echoresponses; (b) generating, by the adaptive filter, a sum of saidestimates; and (c) generating, by the adaptive filter, an error signalrepresenting a difference between said signal and said sum of saidestimates, wherein: said estimates are generated using afrequency-domain recursive least-squares algorithm in which at least oneof said estimates corresponding to a first reception channel is based on(i) an impulse response corresponding to the first reception channel and(ii) one or more other impulse responses corresponding to one or moreother reception channels.
 2. The method set forth in claim 1 whereinsaid estimates are generated by diagonally decomposing by Fouriertransformation a circulant matrix formed by augmentation of a matrix ofvectors representing said input signal.
 3. The method set forth in claim1 wherein said step (a) of generating said estimates comprises, for saidestimates, the steps of: (a1) forming a matrix of vectors representingsaid input signal; (a2) augmenting said matrix to form a circulantmatrix; and (a3) decomposing said circulant matrix by Fouriertransformation to form a diagonal matrix, D.
 4. The method of claim 1,wherein the at least one of said estimates is generated based on anestimate of (1) a power spectrum corresponding to the first receptionchannel and (2) one or more cross-power spectra corresponding to (i) thefirst reception channel and (ii) the one or more other receptionchannels.
 5. The invention of claim 1, wherein: the multiple echoresponses are generated at a first transceiver based on multiplereception signals received from a second transceiver via the firstreception channel and the one or more other reception channels; theadaptive filter is co-located with the first transceiver; and the errorsignal is transmitted from the adaptive filter to the secondtransceiver.
 6. The invention of claim 1, wherein: the multiple echoresponses are generated at a first transceiver based on multipletransmission signals transmitted from a second transceiver via the firstreception channel and the one or more other reception channels; theadaptive filter is co-located with the second transceiver; and thesignal is transmitted from the first transceiver to the adaptive filter.7. The invention of claim 1, wherein the method is implemented in amulti-channel teleconferencing device.
 8. The invention of claim 1,wherein: the multiple echo responses comprise (i) a first echo responsecorresponding to a first reception signal transmitted over a firstreception channel and (ii) one or more other echo responsescorresponding to one or more other reception signals transmitted overone or more other reception channels; and the first reception signal hasa non-linear relationship with the one or more other reception signals.9. The invention of claim 8, further comprising the step of generatingthe non-linear relationship by combining an additive signal componentwith at least one of (i) the first reception signal and (ii) the one ormore other reception signals such that a coherence magnitude between thefirst reception signal and the one or more other reception signals isless than a value of one.
 10. An apparatus comprising an adaptive filterthat filters a signal comprising multiple echo responses, wherein eachof said multiple echo responses corresponds to a different receptionchannel, the adaptive filter comprising: two or more filters, whereineach filter generates an estimate of an echo response corresponding to adifferent one of said multiple echo responses; a summing circuit thatgenerates a sum of said estimates; and a subtraction circuit thatgenerates an error signal representing a difference between said signaland said sum of said estimates, wherein: said two or more filtersgenerate said estimates using a frequency-domain recursive least-squaresalgorithm in which at least one of said estimates corresponding to afirst reception channel is based on (i) an impulse responsecorresponding to the first reception channel and (ii) one or more otherimpulse responses corresponding to one or more other reception channels.11. The apparatus of claim 10, wherein said estimates are generated bydiagonally decomposing by Fourier transformation a circulant matrixformed by augmentation of a matrix of vectors representing said inputsignal.
 12. The apparatus of claim 10, wherein said two or more filtersgenerate said estimates by: forming a matrix of vectors representingsaid input signal; augmenting said matrix to form a circulant matrix;and decomposing said circulant matrix by Fourier transformation to forma diagonal matrix, D.
 13. The invention of claim 10, wherein the atleast one of said estimates is generated based on an estimate of (1) apower spectrum corresponding to said first reception channel and (2) oneor more cross-power spectra corresponding to (i) said first receptionchannel and (ii) said one or more other reception channels.
 14. Theinvention of claim 10, wherein: the multiple echo responses aregenerated at a first transceiver based on multiple reception signalsreceived from a second transceiver via the first reception channel andthe one or more other reception channels; the apparatus furthercomprises the first transceiver; and the error signal is transmittedfrom the adaptive filter to the second transceiver.
 15. The invention ofclaim 10, wherein: the multiple echo responses are generated at a firsttransceiver based on multiple transmission signals transmitted from asecond transceiver via the first reception channel and the one or moreother reception channels; the apparatus further comprises the secondtransceiver; and the signal is transmitted from the first transceiver tothe adaptive filter.
 16. The invention of claim 10, wherein theapparatus is a multi-channel teleconferencing device.
 17. The inventionof claim 10, wherein: the multiple echo responses comprise (i) a firstecho response corresponding to a first reception signal transmitted overa first reception channel and (ii) one or more other echo responsecorresponding to one or more other reception signals transmitted overone or more other reception channels; and the apparatus is part of acommunications system comprising one or more non-linear transformationcomponents that generate a non-linear relationship between (i) the firstreception signal and (ii) the one or more other reception signals. 18.The invention of claim 17, wherein the one or more non-lineartransformation components generate the non-linear relationship bycombining an additive signal component with at least one of (i) thefirst reception signal and (ii) the one or more other reception signalssuch that a coherence magnitude between the first reception signal andthe one or more other reception signals is less than a value of one.